Solve (1+1/2)(1+1/2^2)(1+1/2^4)(1+1/2^8)+frac{1}{2^14}=

$$( 1 + \frac { 1 } { 2 } ) ( 1 + \frac { 1 } { 2 ^ { 2 } } ) ( 1 + \frac { 1 } { 2 ^ { 4 } } ) ( 1 + \frac { 1 } { 2 ^ { 8 } } ) + \frac { 1 } { 2 ^ { 14 } } =$$

Evaluate

$$\frac{65537}{32768}=2.000030518$$

Solution Steps

Convert 1 to fraction \frac{2}{2}.

$$\left(\frac{2}{2}+\frac{1}{2}\right)\left(1+\frac{1}{2^{2}}\right)\left(1+\frac{1}{2^{4}}\right)\left(1+\frac{1}{2^{8}}\right)+\frac{1}{2^{14}}$$

Since \frac{2}{2} and \frac{1}{2} have the same denominator, add them by adding their numerators.

$$\frac{2+1}{2}\left(1+\frac{1}{2^{2}}\right)\left(1+\frac{1}{2^{4}}\right)\left(1+\frac{1}{2^{8}}\right)+\frac{1}{2^{14}}$$

Add 2 and 1 to get 3.

$$\frac{3}{2}\left(1+\frac{1}{2^{2}}\right)\left(1+\frac{1}{2^{4}}\right)\left(1+\frac{1}{2^{8}}\right)+\frac{1}{2^{14}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{3}{2}\left(1+\frac{1}{4}\right)\left(1+\frac{1}{2^{4}}\right)\left(1+\frac{1}{2^{8}}\right)+\frac{1}{2^{14}}$$

Convert 1 to fraction \frac{4}{4}.

$$\frac{3}{2}\left(\frac{4}{4}+\frac{1}{4}\right)\left(1+\frac{1}{2^{4}}\right)\left(1+\frac{1}{2^{8}}\right)+\frac{1}{2^{14}}$$

Since \frac{4}{4} and \frac{1}{4} have the same denominator, add them by adding their numerators.

$$\frac{3}{2}\times \frac{4+1}{4}\left(1+\frac{1}{2^{4}}\right)\left(1+\frac{1}{2^{8}}\right)+\frac{1}{2^{14}}$$

Add 4 and 1 to get 5.

$$\frac{3}{2}\times \frac{5}{4}\left(1+\frac{1}{2^{4}}\right)\left(1+\frac{1}{2^{8}}\right)+\frac{1}{2^{14}}$$

Multiply \frac{3}{2} times \frac{5}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3\times 5}{2\times 4}\left(1+\frac{1}{2^{4}}\right)\left(1+\frac{1}{2^{8}}\right)+\frac{1}{2^{14}}$$

Do the multiplications in the fraction \frac{3\times 5}{2\times 4}.

$$\frac{15}{8}\left(1+\frac{1}{2^{4}}\right)\left(1+\frac{1}{2^{8}}\right)+\frac{1}{2^{14}}$$

Calculate 2 to the power of 4 and get 16.

$$\frac{15}{8}\left(1+\frac{1}{16}\right)\left(1+\frac{1}{2^{8}}\right)+\frac{1}{2^{14}}$$

Convert 1 to fraction \frac{16}{16}.

$$\frac{15}{8}\left(\frac{16}{16}+\frac{1}{16}\right)\left(1+\frac{1}{2^{8}}\right)+\frac{1}{2^{14}}$$

Since \frac{16}{16} and \frac{1}{16} have the same denominator, add them by adding their numerators.

$$\frac{15}{8}\times \frac{16+1}{16}\left(1+\frac{1}{2^{8}}\right)+\frac{1}{2^{14}}$$

Add 16 and 1 to get 17.

$$\frac{15}{8}\times \frac{17}{16}\left(1+\frac{1}{2^{8}}\right)+\frac{1}{2^{14}}$$

Multiply \frac{15}{8} times \frac{17}{16} by multiplying numerator times numerator and denominator times denominator.

$$\frac{15\times 17}{8\times 16}\left(1+\frac{1}{2^{8}}\right)+\frac{1}{2^{14}}$$

Do the multiplications in the fraction \frac{15\times 17}{8\times 16}.

$$\frac{255}{128}\left(1+\frac{1}{2^{8}}\right)+\frac{1}{2^{14}}$$

Calculate 2 to the power of 8 and get 256.

$$\frac{255}{128}\left(1+\frac{1}{256}\right)+\frac{1}{2^{14}}$$

Convert 1 to fraction \frac{256}{256}.

$$\frac{255}{128}\left(\frac{256}{256}+\frac{1}{256}\right)+\frac{1}{2^{14}}$$

Since \frac{256}{256} and \frac{1}{256} have the same denominator, add them by adding their numerators.

$$\frac{255}{128}\times \frac{256+1}{256}+\frac{1}{2^{14}}$$

Add 256 and 1 to get 257.

$$\frac{255}{128}\times \frac{257}{256}+\frac{1}{2^{14}}$$

Multiply \frac{255}{128} times \frac{257}{256} by multiplying numerator times numerator and denominator times denominator.

$$\frac{255\times 257}{128\times 256}+\frac{1}{2^{14}}$$

Do the multiplications in the fraction \frac{255\times 257}{128\times 256}.

$$\frac{65535}{32768}+\frac{1}{2^{14}}$$

Calculate 2 to the power of 14 and get 16384.

$$\frac{65535}{32768}+\frac{1}{16384}$$

Least common multiple of 32768 and 16384 is 32768. Convert \frac{65535}{32768} and \frac{1}{16384} to fractions with denominator 32768.

$$\frac{65535}{32768}+\frac{2}{32768}$$

Since \frac{65535}{32768} and \frac{2}{32768} have the same denominator, add them by adding their numerators.

$$\frac{65535+2}{32768}$$

Add 65535 and 2 to get 65537.

$$\frac{65537}{32768}$$

Factor

$$\frac{65537}{2 ^ {15}} = 2\frac{1}{32768} = 2.000030517578125$$